Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

For a finite collection A=(Ai)i∈I of locally closed sets in Rn, n⩾3, with the sign si=±1 prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the α-Riesz kernel |x−y|α−n,α∈(0,2], over positive vector Radon measures μ=(μi)i∈I such that each μi,i∈I, is carried by Ai and normalized by μ1(Ai)=ai∈(0,∞), while the interaction between μi,i∈I , is determined by the matrix (sisj)i,j∈I∙. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution λξA=(λiA)i∈I (also in the presence of an external field) if we restrict ourselves to μ with μi⩽ξi,i∈I, where the constraint ξ=(ξi)i∈I is properly chosen. We establish the sharpness of the suffcient conditions on the solvability thus obtained, provide descriptions of the weighted vector α-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the λiA,i∈I. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the α-Riesz energy on a set of vector measures associated with A, as well as on the establishment of an intimate relationship between the constrained minimum α-Riesz energy problem and a constrained minimum α-Green energy problem, suitably formulated. The results are illustrated by examples.